Time needed for a transverse wave to propagate on a string

[spaghed87]

Homework Statement

Consider a string of total length L, made up of three segments of equal length. The mass per unit length of the first segment is mu, that of the second is 2*mu, and that of the third mu/4. The third segment is tied to a wall, and the string is stretched by a force of magnitude T_s applied to the first segment; T_s is much greater than the total weight of the string.

Express the time t in terms of L, mu, and T_s. I must use those variables for this answer, no values were given for those variables so they must be included in the answer.

Homework Equations

Velocity=sqrt(T_s/mu) -velocity of string

where, T_s is the tension of the string and mu is its linear denisty. mu=mass/length

The Attempt at a Solution

My answer

time=((1+(1/sqrt(2))+2)*sqrt(T_s/mu))/(L/3)

since, time=v/m ==> m/s/m = seconds right? Edit: = 1/seconds

How I got that:

By pulling out the number multiplied by mu from the sqaure root I get:

The first segment is a velocity of 1*sqrt(T_s/mu)

The second segment is a velocity of 1/sqrt(2)*sqrt(T_s/mu)

The third segment is a velocity of 2*sqrt(T_s/mu)

Add those velocities toegther. Then to get the time you can just divide the velocity by the length of the string that is divided into three equal segments. So, dividing by (L/3) will give the answer. Anyone see what is wrong with my answer?

 

[Redbelly98]

... time=v/m ==> m/s/m = seconds right?

Well, no, those units would be 1/seconds or seconds^-1, so something is wrong.

Your equation says that the faster the wave, the longer it would take, which is wrong intuitively. Faster waves should take less time to travel along the string.

 

[spaghed87]

I was thinking I had 1/s for time instead of s but I was having a brain fart. So, if I take the inverse of that equation it should be right then?

 

[spaghed87]

I ended up solving it differently by finding t_1, t_2, t_3, and then by adding all of the times for the time overall. Woot extra credit for me. Thanks for the help. Time for the test tomorrow.